PHYSICAL REVIEW A 72, 032724 共2005兲
Metastable trapping of low-energy positrons by cubane: A computational experiment
F. A. Gianturco*
Department of Chemistry, University of Rome “La Sapienza” and INFM, Piazzale A. Moro 5, 00185 Rome, Italy
P. Nichols
Department of Chemistry, University of New Orleans, New Orleans, Louisiana 70148, USA
T. L. Gibson
Department of Physics, Texas Tech University, P. O. Box 4180, Lubbock, Texas 79409-1051, USA
R. R. Lucchese
Department of Chemistry, Texas A&M University, College Station, Texas 77843-3255, USA
共Received 11 April 2005; published 21 September 2005兲
Quantum-mechanical calculations of low-energy collisions of positrons with cubane 共C8H8兲 molecules are
carried out using a model interaction potential and solving the corresponding scattering equations. The possible
presence of low-energy resonances is analyzed in detail and one of them is found to reside largely inside the
carbon frame of the cubane. The consequences of this finding are discussed and analyzed.
DOI: 10.1103/PhysRevA.72.032724
PACS number共s兲: 34.10.⫹x, 36.10.Dr
I. INTRODUCTION
The study of low-energy scattering of positrons from
molecular gases and solids has received a great deal of attention in the past few years, both experimentally and theoretically 关1–3兴. This interest has been prompted by the broad
range of applications that have been found for thermal or
subthermal positron beams. Such studies have been facilitated by the marked improvements in the technology that
have provided better sources and better beam capabilities 关4兴
and by the corresponding advances in the computational
models which can describe positron processes involving
large molecular systems 关5兴. Due to the presence of a much
higher density of internal states in molecular systems, molecules provide a more efficient environment than atoms for
slowing down positrons. Furthermore, the positron-molecule
interaction has a richer variety of possible channels compared to when electrons are used as probes of molecular
systems. For example, there is the possibility of forming positronium 共Ps兲 compounds which suggests that molecular ions
can be prepared in a “gentle” manner, i.e., with fairly little
internal excitation of the final cations, and therefore could be
experimentally manipulated for longer times than those
formed by electron impact ionization techniques. Additionally, the particle-antiparticle decay channel, whereby the
pick-off of a bound electron by the positron leads to the
emission of two ␥ photons and again to cation formation
in a molecular gas, is also an additional channel in the case
of positron projectiles which becomes very interesting in a
molecular environment 关3兴.
The analogy with electron scattering can be extended to
complex molecular systems, e.g., benzene or sulfur hexafluo-
*Corresponding author. Email address: fa.gianturco@caspur.it;
FAX: ⫹39-06-49913305.
1050-2947/2005/72共3兲/032724共11兲/$23.00
ride, where the experiments and the calculations for lowenergy collision processes 关6,7兴 have revealed a great variety
of resonant processes which cause marked enhancement in
the corresponding elastic and inelastic cross sections and
which also dominate the subsequent “unimolecular”-type decay with fragmentation of the initial species and frequent
electron attachment to one of the fragments 共for a recent
review see, e.g., Ref. 关8兴兲. In such instances, in fact, the
likely existence of precursor states that correspond to electron trapping by dynamical angular momentum effects
共shape resonances兲 has been suggested in many cases 关5–7兴
and has provided a useful rationalization of a broad variety
of experiments involving low-energy electrons 关9兴.
Because of the differences which exist, at the nanoscopic
level, between the types of forces which drive the abovementioned processes with electrons and the similar dynamics
with slow positrons, it is certainly intriguing to investigate
how many of the possible outcomes from positron scattering
off the more complicated polyatomic gases could be either
explained, or predicted, by considering the dynamical trapping at low energies by the molecule. In recent studies on
simple polyatomics, we have shown that positrons form virtual states with vibrationally excited molecular species
关10,11兴 and that such states have marked consequences for
the features associated with, for example, the annihilation
parameters of the ambient gas.
In the present computational analysis, therefore, we investigate whether a realistic model of the interaction forces between e+ and the cage structure of the cubane molecule depicted in Fig. 1 can lead to the possible trapping of a positron
by the molecule and indicate where within that structure the
positron density will be localized. In the next section, we
outline our theoretical and computational model while we
present our results in Sec. III. Our present conclusions are
given in Sec. IV.
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©2005 The American Physical Society
PHYSICAL REVIEW A 72, 032724 共2005兲
GIANTURCO et al.
adapted angular functions in Eqs. 共2兲 and 共3兲 are defined by
p␮
p␮
Xlh
共r̂兲 = 兺 blmh
Y lm共r̂兲.
共4兲
m
The details about the computation of the transformation map␮
have been given before and will not be repeated
trices blmh
here 关12兴.
We assume that the interaction potential can be written in
a local form Vloc共r p兲 共here the static and correlationpolarization potentials兲 for a positron-molecule collision.
The SCE then results in the following set of radial differential equations:
冋
FIG. 1. Three-dimensional view of the equilibrium structure of
cubane, showing the atoms of the carbon cage and the outer hydrogen atoms.
II. THE THEORETICAL MODEL
In this study, we represent the interaction potential between the impinging positron and the molecular target with a
local form. The potential contains contributions from the
electrostatic interaction of e+ with the molecular nuclei 共kept
fixed at their equilibrium geometry兲 and the bound molecular
electrons plus the correlation effects from the short-range
dynamical couplings between the bound electrons and the
positron, Vcorr, that evolve for the long-range region of large
e+-molecule distances into the dipole polarization potential
Vpol. We shall consider two different correlation-polarization
potentials,
Vpcp = Vcorr + Vpol ,
共1兲
which have been used for polyatomic targets, and the corresponding differences in scattering attributes.
A. Single center expansion equations
In our approach, any three-dimensional function, being
either one of the bound state orbitals ␾i共r兲 for one of the N
bound electrons or the scattering wave function ␺共r p兲 describing the impinging positron, is written as a single-center
expansion 共SCE兲 located at the center of mass of the
molecule,
␾ipi␮i共r兲 =
1
兺 upi␮i共r兲Xlhpi␮i共r̂兲,
r l,h ilh
共2兲
册
d2 li共li + 1兲
p␮
−
+ k2 ␺ijp␮共r p兲 = 2 兺 关Vloc,ik共r p兲␺kj
共r p兲兴 ,
dr2p
r2p
k
共5兲
where indices i , j , k represent pairs of angular channel indices 共l , h兲.
The required full interaction potential in Eq. 共5兲 can be
written as
Vloc共r p兲 = Vst共r p兲 + Vpcp共r p兲,
共6兲
where Vst共r p兲 is the familiar static potential between the impinging positron and the target 共bound electrons+ nuclei兲 and
Vpcp describes the short-range and long-range correlationpolarization interaction forces 关5兴. The SCE equations can
then be solved using standard methods for solving ordinary
differential equations 关13兴.
A rigorous approach for the inclusion of positron-electron
correlation is to use an extensive configuration-interaction
expansion of the target electronic wave function over a suitable set of excited electronic configurations and further improvement of the wave function by adding Hylleraas-type
functions which can describe the positron wave function
within the physical space of the target electronic charge distribution 关14兴. Such expansions, however, are markedly
energy-dependent, and usually converge too slowly to be a
useful tool for a general implementation with complex molecular targets, where truncated expansions need to be very
large before they begin to be realistic in describing correlation effects 关15,16兴.
As a less computationally intensive alternative, we have
developed global models of the correlation-polarization effects which do not depend on empirical parameters but can
be implemented via a simplified, local representation of the
correlation-polarization interactions, Vpcp共r p兲.
B. The distributed positron model
␺ p␮共r p兲 =
1
兺 ␺p␮共rp兲Xlhp␮共r̂p兲,
r p l,h lh
共3兲
where i labels a specific, multicenter occupied orbital, which
belongs to a specific irreducible representation 共IR兲 of the
point group of the molecule. The index p labels a relevant IR
and ␮ indicates one of its components. The index h labels a
specific angular basis function for a given partial wave l,
used within the ␮th component of the pth IR. The symmetry-
One model correlation-polarization potential we have
used to Vpcp in Eq. 共6兲 is the distributed positron model
DPM
关17,18兴. The form adopted here for
共DPM兲 potential, Vpcp
DPM
the Vpcp is based on a modification of the adiabatic polarization effect which makes use of quantum chemistry technology to provide a variational estimate of the polarization
potential. In the adiabatic approximation to that potential, in
fact, the positron is treated as an additional “nucleus” 共a
point charge of +1兲 fixed at location r p with respect to the
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METASTABLE TRAPPING OF LOW-ENERGY POSITRONS…
center of mass of the atomic or molecular target. The target
electronic orbitals are allowed to relax fully in the presence
of this fixed additional charge and the energy lowering due to
the distortion is recorded. This energy lowering represents
the adiabatic polarization potential at one point in space. Of
course, in order to represent fully the spatial dependence of
this interaction, many such points must be computed.
However, due to nonadiabatic and short-range correlation
effects, e.g., virtual Ps formation, the adiabatic approximation can overestimate the strength of the polarization potential for smaller values of r p, where the positron has penetrated the target electronic cloud. The present model
corrects for this by treating the positron as a “smeared out”
distribution of charge rather than as a point charge. If the
scattering particle really were an additional point charge,
then the dominant short-range correlation effect would be
virtual hydrogen atom formation into ground and excited
states, and the ␦-function distribution of positive charge at
the center of mass would be correct. But, for a Ps atom, the
positive charge is not localized at the center of mass, and to
mimic this effect in computing the polarization potential
we represent the positron as a spherically symmetric distribution of charge. This leads to a polarization potential that
more closely reflects the correct physics and that smoothly
reduces to the expected result for larger values of r p without
any need to select a crossover distance or make additional
ad hoc corrections.
Within the DPM one can, in principle, choose any reasonable distribution that approximates the positive charge for
virtual Ps embedded in the near-target environment. However, it is our experience that the results are fairly sensitive to
the effective size of the distribution. In our initial proof-ofconcept studies 关17,18兴, we selected two very simple uniform spherical charge distributions based on the radius of the
positive charge 共with respect to the Ps center of mass兲 of
1.0 bohr, the maximum in the radial distribution, and the
average radius of 1.5 bohr. Both cases provided scattering
results that were in significantly better agreement with measured values than those obtained with the adiabatic polarization potential, with the larger radius distribution being better
for molecular targets. To implement the model for polyatomic target molecules 关19,20兴, we construct the positron
charge distribution from a 1s STO-3G basis function with the
tighter Slater exponent of ␰ = 1.24 that is recommended 关21兴
for a molecular environment. The STO-3G distribution
yields scattering results close to that of the larger uniform
spherical distribution, but is much easier to implement nuDPM
potential is calculated, it is commerically. Once the Vpcp
bined with the static potential to yield the total local interaction potential of Eq. 共6兲.
Subsequent to our development of the DPM to account
for nonadiabatic polarization effects in positron-molecule
scattering, a somewhat similar scheme was proposed by
Bouferguene et al. 关23兴 for low-energy electron-H2 collisions
in which the polarization interaction is computed by replacing the impinging electron with a spherical Gaussian distribution of charge −1. Like the DPM for positron scattering,
this has the effect of reducing the overestimation of the adiabatic potential near the target and can be very efficiently
implemented within a quantum chemistry framework.
FIG. 2. Computed correlation-polarization potential using the
DPM model, VDPM
pcp , shown along three different directions.
However, for electron scattering, the overestimation of the
adiabatic polarization potential arises from an increase in the
local kinetic energy of the scattering electron when near the
target 关22兴 to a value that invalidates the adiabatic assumption that the rapidly moving target electrons adjust fully to
the presence of a slow projectile electron. Thus, rather than a
single fixed distribution, Bouferguene et al. 关23兴 found it
necessary to use a distribution that varies with the distance of
the scattering electron from the molecular center of mass and
involves a semiempirical parameter. In extending this
method to electron-N2 scattering, Feng et al. 关24兴 had to
introduce additional ad hoc dependences on the orientation
of the target nuclei with respect to the projectile and on the
internuclear separation distance.
In contrast, we have been able to obtain good agreement
with various measured positron-molecule scattering results
within the DPM using essentially the same procedure 共i.e.,
without needing to adjust for details of the target molecule兲
on a variety of systems 关17–20兴 as diverse as H2 and SF6.
This past success is the chief motivation for including the
DPM in the current study.
To implement the model for polyatomic target molecules
关19,20兴, we construct the charge distribution from a 1s
STO-3G basis function with Slater exponent ␰ = 1.24. Once
DPM
potential is calculated, it is combined with the
the Vpcp
static potential to yield the total local interaction potential of
Eq. 共6兲.
C. The DFT correlation-polarization model
DFT
A second model for Vpcp is the Vpcp
potential, which has
been applied to positron scattering earlier 关25兴 and is based
on the correlation energy ␧e-p of a localized positron in an
electron gas. The quantity ␧e-p was originally derived by Arponen and Pajanne 关26,27兴 from the theory that the incoming
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GIANTURCO et al.
FIG. 3. Computed dominant adiabatic potentials for positron
scattering from cubane with the A1g 共upper panel兲 and T1u 共lower
panel兲 IR’s.
FIG. 4. Same as in Fig. 3 but for the Eg 共upper panel兲 and T2g
共lower panel兲 IR’s of the octahedral symmetry of the scattering
potential.
positron is assumed to be a charged impurity at each fixed
distance r p in a homogeneous electron gas, which is in turn
treated as a set of interacting bosons which represent the
collective excitations provided by the random-phase approximation. Based on their work, Boronski and Nieminen 关28兴
gave the interpolation formulae of ␧e-p over the entire range
of the density parameter rs which satisfies the relationship of
4
3
3 ␲rs ␳共r兲 = 1. The relationship between the short-range correlation potential, Vcorr, and ␧e-p, which is consistent with
the local-density approximation and a variational principle
for a total collision system with the size of the target, is
given by 关29兴
by the well-known second-order perturbation expansion formula 共in atomic units兲,
DFT
共r p兲 =
Vcorr
d
兵␳共r p兲␧e−p关␳共r p兲兴其,
d␳
共7兲
where ␳ denotes the undistorted electronic density of the
target. This quantity provides the probability for finding any
of the electrons near the impinging positron.
DFT
potential, however, does not have the correct
The Vcorr
asymptotic form as r p → ⬁, which is important for lowenergy scattering. We note that the asymptotic form of the
interaction is independent of the sign of the impinging
charged particle and, in its simpler spherical form, is given
⬁
␣
l
− 2l+2 ,
兺
2r
r →⬁ l=1
Vpol共r p兲 ⬃
p
共8兲
p
where r p represents the distance from the center of mass of
the molecule, and ␣l are the multipolar static polarizabilities
of the molecule. In most cases, only the lowest order is kept
in the expansion above, and therefore the target distortion is
viewed as chiefly resulting from the induced dipole contribution with the molecular dipole polarizability as its coefficient. The drawback of the above expansion, however, is that
it fails to represent correctly the true short-range behavior of
the full interaction and does not contain any contributions
from dynamic correlation. Therefore, in order to correct for
such failures, one needs to model the dynamical correlation
effects that dominate the short-range behavior by a potential
DFT
given in Eq. 共7兲.
such as Vcorr
We therefore describe the full Vpcp共r p兲 interaction as given
by two distinct contributions which are connected at a
distance, rcp,
032724-4
DFT
DFT
Vpcp
共r p兲 = Vcorr
共r p兲
for r p ⬍ rcp ,
PHYSICAL REVIEW A 72, 032724 共2005兲
METASTABLE TRAPPING OF LOW-ENERGY POSITRONS…
FIG. 5. Computed partial cross sections for all the IR’s that make a significant contribution to the total elastic cross sections.
=Vpol共r p兲
for r p ⬎ rcp .
共9兲
In the present case, the value for rcp is around 4.1 a.u.
DFT
, as defined in
The total local interaction potential, Vloc
Eq. 共6兲, is then given by the sum of the exact static interaction between the impinging positron and the components of
the molecular target, Vst 共for detail forms, see, for example,
DFT
.
关12兴兲, and the Vpcp
It is difficult at this stage to decide which of the two
models is likely to be more realistic. The past record of the
DFT modeling on polyatomic targets has been more extensive because it is easier to generate in terms of computational
cost and has provided thus far good accord with available
experiments 共e.g., see Refs. 关10,11兴兲. On the other hand, the
DPM approach is physically more appealing and is being
tested increasingly more often with available experiments
with reasonable success 共e.g., see Refs. 关19,20兴兲.
III. RESULTS
The calculations were carried out largely following the
numerical details already reported in our previous work on
cubane 关9兴. The molecular structure was taken at its equilibrium geometry of the octahedral structure, using 1.078 Å for
the eight C u H bonds and 1.557 Å for the twelve C u C
bonds. A pictorial view of the 3D structure is shown for
clarity in Fig. 1.
The single-determinant 共SD兲 electronic structure calculations were carried out, as in Ref. 关9兴, using a cc-pVTZ expansion, producing a spherical polarizability of 70.321 a.u.3.
The anisotropic contributions to the long-range potential in
Eq. 共9兲 were generated by equally partitioning that value
over the eight carbon centers and then expanding the overall
polarization potential over the center of mass of the above
structure. The Hartree-Fock 共HF兲 orbitals were also expanded up to lmax = 40 multipolar terms and the ensuing static
and correlation contributions to Eq. 共6兲 were expanded up to
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GIANTURCO et al.
FIG. 6. Computed eigenphase sum and cross
sections below 10 eV for the A1g and T1u partial
contributions.
␭max = 80. The range of radial integration depended on the
collision energy considered and went from about 95 a.u. at
the higher collision energy of 20– 40 eV to more than
103 a.u. for the lower energies of a few meV.
The chief aim of the present study is to see if the simulation of positron scattering from such a cage structure could
lead to the presence of trapped, metastable positron states
inside the carbon network because of the special interplay of
attractive-repulsive contributions of the full interaction with
this molecule. Our earlier study on the C60 carbon cage 关5兴
has indeed suggested that such resonant states exist in the
low-energy region, due to barrier trapping generated by the
Coulomb potential of the carbon nuclei forming the fullerene
structure. It would be of interest to see if the present, smaller,
cage structure could lead to a similar effect. The spatial location of the carbon atoms from the center of mass is, in this
case, at about 2.0 a.u. and we shall therefore see that the
overall size of the present “cage” plays a significant role in
the possible trapping of a positron.
DPM
In Fig. 2, we report the behavior of our computed Vpcp
potential along three distinct directions of approach to the
target molecule. It is interesting to note that, as expected, the
strongest attractive well occurs as the impinging projectile
follows the C u H bond and therefore we see a marked local
correlation distortion of the hydrogen atom around 5 a.u. followed by the stronger carbon atom correlation contribution
with its maximum effect just below 3.0 a.u. On the other
hand, the other two directions of approach, where no atomic
centers are encountered by the impinging positron, clearly
show only the long-range attractive features of the large molecular polarizability of the system that goes to a constant
value near the center of the cage.
A more transparent way of analyzing the spatial features
of the positron-molecule interaction, especially in the case of
highly symmetric systems as in the present instance, is to
employ an adiabatic angular basis set expansion of the form
p␮
Zkp␮共r p兲 = 兺 Xhl
共r̂ p兲Ck,hl共r p兲,
共10兲
hl
where the expansion coefficients Ck,hl共r p兲 are given by a matrix eigenvalue equation
Vhhl⬘l⬘共r p兲Ck,hl共r p兲 = Vk共r p兲Ck,h⬘l⬘共r p兲.
兺
hl
共11兲
The adiabatic eigenvalues Vk therefore depend on the
positron-molecule distance and provide a set of adiabatic potentials for each irreducible representation 共IR兲 of the molecular point group. The nonadiabatic coupling terms can be
included when the coupled radial equations are solved 共see,
for details, our earlier work 关30–32兴兲 leading to the same
results as would be obtained if one solved the usual SCE
equations. The adiabatic potentials can provide a fairly compact description of the dominant features of the positronmolecule interaction.
Figures 3 and 4 report the dominant multipolar components for the adiabatic representations of e+-cubane interaction for four of the dominant IR’s associated with the equilibrium geometry, which has octahedral symmetry.
The calculations reported in the two figures clearly show
the dominant partial waves, which are active within the
range of energies of our present study, i.e., roughly the region from threshold up to about 30 eV. The following considerations could be made by inspecting the potential features reported there.
共i兲 Essentially all adiabatic potentials turn out to be dominated by repulsive forces 共nuclear static contributions plus
centrifugal terms兲 as the positron approaches the center of
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METASTABLE TRAPPING OF LOW-ENERGY POSITRONS…
FIG. 7. Computed eigenphase sums for the same symmetry contributions to the cross section reported in Fig. 5.
the cage, with only a very weak indication of long-range
attractive features clearly located outside the carbon cage.
共ii兲 The totally symmetric component A1g is the only adiabatic interaction with an s-wave contribution which gives
rise to a clear barrier at the boundary of the carbon atom
network. This is obviously not a dynamical effect from angular momentum conservation but rather a feature related to
the spatial properties of the full interaction that is a balance
of the attractive Vpcp potential with the strongly repulsive Vst
interaction of Eq. 共6兲.
We therefore expect that the A1g symmetry component
might be the most likely contribution to the total elastic cross
section which could provide evidence for metastable trapping of low-energy positrons by the carbon cage of cubane.
In Fig. 5, we report all the computed components of
the total elastic 共rotationally summed兲 cross sections using
DPM
interaction described in the previous section. It
the Vpcp
is interesting to see that the largest cross-section contributions come, at low energies, from the T1u, T2g, Eg, and
A1g components and that there are several cross-section
peaks over the energy range we have examined that
also correspond to a marked rising behavior in their
eigenphase sums, as discussed later. The most notable indicators of resonant behavior can be seen from the strong
peaks, in Fig. 6, of the A1g and T1u cross sections which
correspond to steady and strong increases 共around the same
energies兲 of their corresponding eigenphase sums. They further show, at such energies, poles of the S matrix in the
complex plane, from which resonance properties can be
extracted.
To analyze such components in greater detail, we report in
Fig. 6 the low-energy behavior of the A1g and T1u contributions, showing in the left panels the eigenphase sums and in
the right panels the corresponding cross sections. The following considerations could be made.
共i兲 The partial cross section for the A1g contribution is
finite and large at zero energy and falls monotonically down
to the high-energy resonance, while the structure seen at low
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GIANTURCO et al.
FIG. 9. Three-dimensional view of the A1g resonant orbital with
superimposed carbon cage structure of cubane.
FIG. 8. Computed resonant wave functions for the two S-matrix
poles of A1g 共upper兲 and T1u 共lower panel兲 symmetries. Only the
dominant partial-wave components are shown.
energy in its cross section is due to the eigenphase sum going
through a multiple of ␲ around 1 eV.
共ii兲 By analyzing the low-energy behavior of the dominant
l = 0 phase shift 共eigenphase sum兲, one obtains the corresponding scattering length of 43.62 bohr. Since it is a positive quantity, there is no virtual state for positron scattering
at vanishing collision energies.
共iii兲 The T1u component shows a marked increase of the
eigenphase sum around 1 eV and a corresponding increase
of the cross section. As we shall show later on, it corresponds
to an “above-the-barrier” resonant behavior with negligible
positron localization.
The results of the eigenphase sum calculations for all the
contributing partial symmetries are shown by Fig. 7.
The corresponding scattering wave functions are shown in
Fig. 8, where we report the dominant component of the A1g
resonance in the upper panel and the dominant partial wave
for the T1u resonance in the lower panel. The results of this
figure are rather interesting in that they clearly show the
presence of an e+-C8H8 complex with a strong amplitude
contribution inside the C cage and supported at around
23 eV by the potential barrier seen in Fig. 3. One also sees,
however, that the corresponding widths from our fixed-nuclei
calculation are really very broad. Although one should expect that the inclusion of nuclear dynamics may markedly
modify the features of this metastable complex, it is difficult
to imagine that such a large amount of excess energy could
be efficiently transferred to the carbon cage without also
opening up molecular breakup channels.
A pictorial, three-dimensional view of the corresponding
resonant orbital is reported in Fig. 9, where the carbonhydrogen cage structure is shown with the wave function:
it is clear from the results presented in Figs. 8 and 9 that
the resonant state does correspond to trapping the positron
inside the cage, although both width and energy position
do not cause this resonance to be physically very likely to
occur because of the large amount of positive energy which
needs to be disposed of during the very short lifetime of the
metastable state.
The low-energy resonance observed for the T1u symmetry
共lower panel of Fig. 8兲 is, on the other hand, much narrower
and corresponds to a longer lifetime for the trapped positron.
The resonant wave function shown in that figure, however,
indicates this state to be dominated by p-wave dynamical
trapping of the scattered positron well outside the carbon
cage. It is therefore seen as a very diffuse shape resonance
with the positron density distributed around the cage structure and well outside it. As discussed before, since we are
dealing with a resonance state above the barrier, the corresponding scattering wave function is still very diffuse and
will be very difficult to represent as localized within the carbon cage structure.
In the upper panels of Fig. 10, we further report the behavior of the total cross sections obtained as a sum of the
contributions already shown by Fig. 5 and computed by using both the DPM and DFT potentials discussed in this work.
The log scale for the cross sections in the upper right-hand
panel helps us to better see the very low-energy behavior of
such cross section and the very marked increase of this value
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METASTABLE TRAPPING OF LOW-ENERGY POSITRONS…
FIG. 10. Computed total cross sections 共upper panels兲 obtained using the model interactions of the present work. Solid lines: calculations
DFT
via the VDPM
pcp ; dashes: calculations via the Vpcp potential. Lower panels: same comparison for cross section 共left兲 and the eigenphase sum
共right兲 for the totally symmetric contribution, A1g.
DFT
as the collision energy decreases. We also see that the Vpcp
共given there by the dashed curves labeled DFT兲, which is the
stronger of the two potentials, provides larger cross sections
but does not change the general energy dependence of the
cross sections. This is also confirmed by the results reported
in the lower two panels, where we compare the cross sections and the eigenphase sums computed using both potential
models for the A1g IR contribution.
We see in the lower panel, in fact, that the shoulder
around the resonance position is given by both calculations
and so is the Ramsauer-Townsend minimum, although there
are shifts in their location. In a similar fashion, the resonance
feature of the eigenphase sum is moved to lower energies by
the stronger DFT model but it remains present in the A1g
symmetry.
A quantitative comparison between two specific cuts of
the positron-cubane interaction is reported by Fig. 11, where
the direction along the C u H bond is shown for both of
DFT
is clearly much stronger in the short-range
them: the Vpcp
region, an area of interaction which is sampled by the lower
partial waves which dominate the scattering at the lower collision energies: hence the cross-section behavior of Fig. 10.
As mentioned before, it is difficult to decide which of the
two potentials is more likely realistically to provide the correct interaction, although the DPM is physically more appealing and may be able in the future also to yield competitive agreement with scattering observables.
IV. GENERAL CONCLUSIONS
In this work, we have analyzed in some detail whether it
would be physically possible to form temporary, metastable
states of positrons trapped inside the carbon cage provided
FIG. 11. Comparison of the computed Vpcp model potentials of
the present work. The direction shown is the one along the C u H
bonds from the center of mass of the cage.
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GIANTURCO et al.
by the octahedral structure of the cubane 共C8H8兲 molecule in
its isolated form. The motivation, besides the fundamental
interest in studying the possible existence of positronmolecule complex formation in the gas phase, stems also
from the potential interest related to the construction of molecular devices that can be employed to store positrons for
fairly long times. In particular, carbon atom networks where
the net charge inside the closed structures 共“cages”兲 is usually zero can be thought of as possible nanocontainers,
whereby final bound states could be attained after complex
stabilization through the extra-energy dissipation within the
nuclear vibrational modes.
The system that we have studied here, however, has been
considered in a preliminary stage where the nuclear motion
is decoupled from the scattered positron, and therefore the
chief aim here was to establish whether our theoretical models of the forces at play can suggest the presence of such
metastable, trapped-positron states at relatively low energies.
We have considered two different ways in which the
short-range electron-positron correlation-polarization effects,
Vpcp共r p兲, could be included within our quantum dynamical
calculations: the DPM and DFT models discussed in Sec. II.
Although both models were once compared in the past by us
关19,20兴 and have provided very similar results for elastic
integral and differential cross sections for small polyatomics,
we feel that to describe the positron “cloud” as a spatial
entity rather than a pointlike structure is possibly a more
realistic model. In the present study, we have therefore focused mainly on the results from the DPM calculations and
only compared some of the final observables with those obDFT
model potential. It turned out
tained when using the Vpcp
that both model treatments, although differing in the details,
essentially provide the same physical picture for the process
we are considering, i.e., they both indicate positron trapping
inside the cage not to be very likely to occur.
The search for a trapped state has shown in fact that,
although the positron-molecule interaction is essentially repulsive in the short and intermediate distances, and that
fairly shallow attractive regions exist only far away from the
carbon cage of the cubane, the totally symmetric component
of that interaction, the A1g IR, may be capable of providing a
Coulomb barrier located in the region of the carbon cage and
that the impinging positron can be trapped behind that barrier. It would, however, give rise to a broad metastable state
at a fairly high-energy region around 20 eV. If a more realistic dynamical coupling between such a state and the
nuclear vibrations could possibly be included, one would further expect that some of the metastable complex structures
could decay into a finally bound state of the positron, although the competition with the cage break-up channels
would also be very strong. In any event, if that stabilization
channel were to survive, it could be providing a sort of nanoscopic molecular mechanism for such a positron trapping
process to occur.
Another interesting property of the dynamics of positron
scattering off cubane which could be compared with the behavior of electrons is the value of its scattering length, A0,
which could be obtained from 关33兴
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ACKNOWLEDGMENTS
F.A.G. is grateful to the CASPUR supercomputing center
for the computational support and the European Network
EPIC 共HPRN-CT-2002-00286兲 for financial support. F.A.G.
and R.R.L. also acknowledge the NATO Collaborative Research Grant 共2002–2005兲. R.R.L. wishes to acknowledge
the support of the Welch Foundation 共Houston, TX兲 for support under Grant No. A-1020.
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PHYSICAL REVIEW A 72, 032724 共2005兲
METASTABLE TRAPPING OF LOW-ENERGY POSITRONS…
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